Eshelbian dislocation mechanics: J-, M-, and L-integrals of straight dislocations

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In this work, using the framework of (three-dimensional) Eshelbian dislocation mechanics, we derive the J-, M-, and L-integrals of a single (edge and screw) dislocation in isotropic elasticity as a limit of the J-, M-, and L-integrals between two straight dislocations as they have recently been derived by Agiasofitou and Lazar [Int. J. Eng. Sci. 114 (2017) 16–40]. Special attention is focused on the M-integral. The M-integral of a single dislocation in anisotropic elasticity is also derived. The obtained results reveal the physical interpretation of the M-integral (per unit length) of a single dislocation as the total energy of the dislocation which is the sum of the self-energy (per unit length) of the dislocation and the dislocation core energy (per unit length). The latter can be identified with the work produced by the Peach–Koehler force. It is shown that the dislocation core energy (per unit length) is twice the corresponding pre-logarithmic energy factor. This result is valid in isotropic as well as in anisotropic elasticity. The only difference lies on the pre-logarithmic energy factor which is more complex in anisotropic elasticity due to the anisotropic energy coefficient tensor which captures the anisotropy of the material.